Time-Dependent Modeling of Global Electric Circuit Processes
Sotirios A. Mallios and Victor P. Pasko
Communications and Space Sciences Laboratory
Dept. of Electrical Engineering
The Pennsylvania State University
————————
FESD “Electrical Connections and Consequences Within the Earth System”
Meeting, February 28, 2013
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Results
Conclusions
2 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Model Formulation
I
A cylindrical two-dimensional coordinate system (r , z) is used, with the
z− axis representing the altitude.
I
The ground (z0 =0 km), upper (zmax =40 km) and the side (r =80 km)
boundaries are assumed to be perfectly conducting.
I
The spatial distributions of positive and negative source charges are
considered to be Gaussian of the following form:
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Results
Conclusions
ρ± (r , z) =
r2
(z − h± )2
Q±
−
exp
−
(2π)3/2 αz α2r
2α2z
2α2r
where αz = 1 km and αr = 1 km are the vertical and horizontal scales of
the charge distributions, h± are the altitudes of their centers and
Q± = ±1 C are the total charge values that are deposited/removed from
the system
I
The continuous thundercloud charge distribution dynamics can be
described as follows:
I
Linear in time charge accumulation for a time interval equal to 400 sec.
I
Charge removal during a 400 msec time interval due to lightning.
I
Relaxation of the system for a time interval equal to 400 sec.
I
Cloud relaxation for a time interval equal to 450 sec during which the
conductivity inside the thundercloud is increased linearly in time to its ambient
value.
3 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Model Formulation (cont.)
I
Sotirios A. Mallios and
Victor P. Pasko
The induced charge, the potential and the electric field are calculated by
solving the following set of equations:
∇2 φ = −
FESD Meeting 2013
Introduction
Model Formulation
ρt
0
(1)
ρt
∂ρt
− ∇σ · ∇φ = −σ
∂t
0
~ = −∇φ
E
Results
Conclusions
σ(r , z) = σ0 ez/l
1−
c
1 − tanh( r −r
)
α
2
×
(2)
(3)
c
1 − tanh( z−z
)
α
2
!
(4)
where ρt = ρs + ρf the total charge density, σ0 = 5 × 10−14 S/m is the
conductivity at ground level, l = 6 km is the altitude scaling factor, zc = 20
km and rc = 20 km are the vertical and horizontal extents of the cloud and
α = 800 m is the thickness of the conductivity transition region between
the cloud and the surrounding air.
I
The charges that are transferred to the ionosphere and to the ground are
calculated by integrating the conduction current over horizontal planes,
corresponting to the lower and upper boundaries.
I
The total efficiency of a lightning flash as a contributor to the GEC is
defined as the ratio of the total charge transfered to the Ionosphere over
the absolute value of the maximum charge that is accumulated in the
thundercloud [Mallios and Pasko, JGR, 117, A08303, 2012].
4 / 25
Negative Cloud-to-Ground Lightning (-CG)
Charge density magnitude (C/m3)
FESD Meeting 2013
Introduction
Model Formulation
10-12
10-13
Screening
Charges
30
- - -
10
30
20
10
0
10
20
10
30
40
0
40
40
30
30
t=800 sec
0
40
30
20
- - -
10
+
10
0
30
20
10
20
- - -
20
+
t=400.4 sec
40
- - -
Altitude (km)
0
40
- - -
20
+
-
t=400 sec
10-16
- - - - -
30
- - -
20
- - -
Altitude (km)
Conclusions
10-15
40
40
Results
10-14
10
20
Radial Distance (km)
0
10
20
30
40
10
20
30
40
- -
10
+
t=1250 sec
30
- - -
Sotirios A. Mallios and
Victor P. Pasko
- - -
Time-Dependent
Modeling of Global
Electric Circuit
Processes
0
40 40
30
20
+ + +
10
0
Radial Distance (km)
5 / 25
Negative Cloud-to-Ground Lightning (cont.)
Sotirios A. Mallios and
Victor P. Pasko
Model Formulation
Results
Conclusions
0.4
Positive Q
0.5
0
Negative Q
−0.5
−1
0
Induced Q (C)
Introduction
0.6
1
Source Q (C)
FESD Meeting 2013
500
0
−0.2
−0.4
Total Induced Q
−0.6
-0.27
-0.62
−0.8
(b)
Negative Q
−1.2
0
1000
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1 (c)
−1.2
0
Positive Q
0.2
−1
(a)
Deposited Q (C)
Time-Dependent
Modeling of Global
Electric Circuit
Processes
500
-0.98
1000
0.81
Ionospheric Q
0.55
0.27
Ground Q
-0.83
-0.95
500
Time (sec)
1000
6 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Intra-Cloud Lightning (IC)
Sotirios A. Mallios and
Victor P. Pasko
Charge density magnitude (C/m3)
FESD Meeting 2013
10-12
10-13
10-14
10-15
10-16
Introduction
40
40
Screening
Charges
Conclusions
- - +
-
- - -
20
10
t=400 sec
30
20
10
0
10
20
10
t=400.4 sec
30
40
0
40
40
30
30
+++
- - -
10
t=800 sec
0
40
30
20
10
0
30
20
10
0
10
20
30
10
0
10
20
30
40
20
- +
- ++
20
- - -
20
40
+++
- - -
Altitude (km)
0
40
+ + + + +
30
- - -
Altitude (km)
30
- - -
Results
- - -
Model Formulation
10
20
Radial Distance (km)
10
t=1250 sec
30
40
0
40
30
20
Radial Distance (km)
40
7 / 25
Intra-Cloud Lightning (cont.)
Introduction
Model Formulation
Results
Conclusions
Source Q (C)
FESD Meeting 2013
0.2
1
Positive Q
0.5
Induced Q (C)
Sotirios A. Mallios and
Victor P. Pasko
0
0
500
0.04
0
-0.04
Total
induced Q
−0.2
(a)
−1
Negative Q
−0.3
0
1000
0.18
Positive Q
0.1
−0.1
Negative Q
−0.5
500
0.3
Deposited Q (C)
Time-Dependent
Modeling of Global
Electric Circuit
Processes
(b)
1000
-0.28
0.27
0.2
Ionospheric Q
0.1
0
−0.1
0
(c)
Ground Q
500
Time (sec)
1000
8 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Electrified cloud that does not produce lightning
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
0.6
Introduction
Model Formulation
Results
Deposited Q (C)
Conclusions
0.4
0.25
0.2
Ionospheric Q
0
−0.2
Ground Q
−0.4
−0.6
0
500
Time (sec)
-0.25
1000
9 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Study of a Realistic Thunderstorm Configuration
Charge density magnitude (C/m3)
Sotirios A. Mallios and
Victor P. Pasko
40
10-9
10-10
10-11
10-12
10-13
Introduction
Model Formulation
Results
Conclusions
Altitude (km)
FESD Meeting 2013
30
Screening
Charges
20
- - - -
10
++++
6667 C
- - - -
-10000 C
3333 C
++++
0
- -
200
100
- -
0
100
Radial Distance (km)
200
I
Mach et al. [JGR, 114, D10204, 2009; JGR, 115, D03201, 2010; JGR,
116, D05201, 2011] presented analyses of high-altitude aircraft
observations regarding the Wilson currents that flow above thunderstorms
and electrified clouds during 850 overflights.
I
We reproduce the thundercloud configuration for one of these
thunderstorms, knowing the Wilson currents that were estimated 20 km
above the thunderstorm as well as the altitudes of the charge layers inside
the thundercloud.
10 / 25
Study of a Realistic Thunderstorm Configuration (Cont.)
40
1
30
Einit
Einit
Altitude (km)
FESD Meeting 2013
Depostited Q (C)
Sotirios A. Mallios and
Victor P. Pasko
20
Introduction
Model Formulation
Results
10
Conclusions
0
3
Transfered Q (C)
Positive
0
−0.5
(a)
(b)
−2
−1
4
0
Ez (kV/cm)
1
−1
0
2
x 10
Negative
1000
2000
Time (sec)
3000
6
Ionospheric
2
4
1
2
0
Ionosphere and at 20 km
0
−2
−1
−2
5
x 10
0.5
I (A)
Time-Dependent
Modeling of Global
Electric Circuit
Processes
−3
0
Ground
−4
Ground
−6 (d)
(c)
1000
2000
Time (sec)
3000
0
1000
2000
Time (sec)
3000
I
Out of 82570 C that are deposited to the thundercloud, 22350 C are
transfered to the ionosphere, which are 27.07% of the deposited charge.
I
Above the thundercloud the current is the same at any altitude in the
steady state.
11 / 25
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Results
Conclusions
Study of a Realistic Thunderstorm Configuration (Cont.)
0.6
0.6
h = 2 km
0.5
Deposited Q (C)
Time-Dependent
Modeling of Global
Electric Circuit
Processes
0.4
0.4
Ionospheric Q
0.3
0.2
0.2
0
7
8
9
10
h+ (km)
11
Ground Q
0.1
(a)
−0.1
6
Ionospheric Q
0.3
Ground Q
0.1
0
h+ = 13 km
0.5
12
13
(b)
−0.1
2
3
4
5
6
h− (km)
7
8
9
I
In [Mallios and Pasko, JGR, 117, A08303, 2012] we have shown the
dependancy between the charge that is transfered to the ionosphere and
the distance between the source charges, for the case of a dipole during
the electrification stage of a thunderstorm.
I
For a distance equal to 5 km, 26% of the deposited charge is transfered to
the ionosphere. This value has only 5% difference from the value that we
derived for the case of a realistic thunderstorm.
I
This shows the validity of our model, and that the main parameters that
influence the amount of the charge that is transfered to the ionosphere
during the electrification phase of a thunderstorm are only the distance
between the main positive and main negative charge layers, as well as the
amount of the deposited charge.
12 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Sotirios A. Mallios and
Victor P. Pasko
Corona Current
I
When the electric field near the ground exceeds a few kilovolts per meter,
ions are produced by corona and flow upward from the ground to the base
of the thundercloud.
I
The minimum electric field required to produce corona is about 5 kV/m
depending upon the surface roughness and vegetation growth [Standler
R. B., JGR, 85, 4541, 1980].
I
When the electric field at the ground exceeds the corona threshold Ec = 5
kV/m we calculate the corona current density from the formulas:
Ez0
−1
(5)
Jc0 = σ0 Ez0 − σ0 Ec exp
Ec
z
Jc (z) = Jc0 exp −
(6)
hl /10
FESD Meeting 2013
Introduction
Model Formulation
Results
Conclusions
where hl is the altitude of the lower boundary of the thundercloud, σ0 is
the value of the conductivity at the ground, and Ez0 the value of the
vertical component of the electric field at the ground.
I
From the corona current density, we calculate the corona current and we
add this current source to the current in continuity equation (2).
13 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Corona Current (Cont.)
Charge density magnitude (C/m3)
Sotirios A. Mallios and
Victor P. Pasko
40
10-9
10-10
10-11
10-12
10-13
FESD Meeting 2013
Results
Conclusions
30
Screening
Charges
20
- - - ++++
10
- - - - -
0
40
Altitude (km)
Model Formulation
Altitude (km)
Introduction
30
- -
Screening
Charges
20
- - - -
Corona
Charges
++++
10
0
++++
- - - -
- 200
100
+
++++
- +
- 0
Radial Distance (km)
100
200
14 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Corona Current (Cont.)
Sotirios A. Mallios and
Victor P. Pasko
40
3
(a)
FESD Meeting 2013
Einit
Einit
Transfered Q (C)
30
20
0
Without
Corona
−2
−1
0
E (kV/cm)
z
2
10
With
Corona
Jc (nA/m )
Conclusions
Altitude (km)
Results
1
1
0
−1
−2
Ground without
corona
−3
0
2
(b)
Ionospheric
2
Introduction
Model Formulation
4
x 10
1000
0
Ground
with corona
2000
Time (sec)
3000
(c)
−5
−10
−15
0
5
10
E (kV/m)
z
15
20
15 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Time-Dependent Global Electric Circuit Model
Sotirios A. Mallios and
Victor P. Pasko
I
We place a positive source charge Q+ at h+ = 10 km and a negative
source charge Q− at h− = 5 km.
I
The vertical and horizontal scales of the source charges are αz = 2 km
and αr = 2 km respectively.
I
We charge the system for a time period of 1000 sec with source currents
I± = 1 A.
I
The ambient conductivity profile distribution is chosen to be:
FESD Meeting 2013
Introduction
Model Formulation
Results
Conclusions
σ(z) = σ0 ez/l
(7)
10−14
where σ0 = 5 ×
S/m is the conductivity at ground level, l = 6 km is
the altitude scaling factor.
I
The boundary conditions are:
I
I
I
Φ = 0 at z = 0 km
Ez = 0 at z = 60 km
Er = 0 at r = 0 km and r = 1000 km.
16 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Time-Dependent Global Electric Circuit Model (Cont.)
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Charge density magnitude (C/m3)
Results
Conclusions
10-10
10-11
10-12
10-13
10-14
60
Altitude (km)
50
40
30
20
+
10
0
1000
-
500
0
32 C
-74 C
Radial Distance (km)
500
1000
17 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Time-Dependent Global Electric Circuit Model (Cont.)
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Results
Static Model
Conclusions
60
Time Dynamic Model
60
50
Altitude (km)
50
40
40
30
30
20
20
10
10
(a)
0 −2
10
0
2
4
10
10
10
Thundercloud Ez Field (V/m)
10
6
(b)
0 −2
10
0
2
4
10
10
10
Thundercloud Ez Field (V/m)
10
18 / 25
6
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Time-Dependent Global Electric Circuit Model (Cont.)
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Results
Static Model
4
3 x 10
Conclusions
3
Potential(V)
2.5
Time Dynamic Model
2.5
2
2
1.5
1.5
1
1
0.5
0
4
x 10
0.5
(a)
0
200
400
600
800
Radial Distance (km)
1000
(b)
0
0
200
400
600
800
Radial Distance (km)
1000
19 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Time-Dependent Global Electric Circuit Model (Cont.)
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Results
60
Altitude (km)
Conclusions
Static Model
60
50
50
40
40
30
30
20
20
10
0
0
Time Dynamic Model
10
(a)
0
2000 4000 6000 8000 10000 0
Fair Weather Potential (V)
(b)
2000 4000 6000 8000
Fair Weather Potential (V)
10000
20 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Time-Dependent Global Electric Circuit Model (Cont.)
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Results
Static Model
60
Altitude (km)
Conclusions
60
50
50
40
40
30
30
20
20
Time Dynamic Model
10
10
0
−1.4 −1.2 −1
(a)
−0.8 −0.6 −0.4 −0.2
Fair Weather Ez Field (V/m)
0
0 −1.4 −1.2
(b)
−1
−0.8 −0.6 −0.4 −0.2
Fair Weather Ez Field (V/m)
0
21 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Time-Dependent Global Electric Circuit Model (Cont.)
Sotirios A. Mallios and
Victor P. Pasko
3
0.4
2.5
FESD Meeting 2013
Potential(V)
0.2
Introduction
Conclusions
I (A)
Model Formulation
Results
0
−0.2
(a)
−0.4
0
4
x 10
2
1.5
1
0.5
200
400
600
Time (sec)
800
1000
(b)
0
0
200
400
600
800
Radial Distance (km)
1000
I
The resistance of the simulation domain is equal to R=38195 Ω.
I
The steady state positive current that flows to the ground is equal to
I=0.2586 A.
I
From Ohm’s law we find that V =9877.2 V.
I
Integrating the potential distribution at the top boundary over the surface
and dividing by the total surface area of the boundary, we find that the
effective potential is equal to <V >=9810 V. This value is consistent with
the value calculated by Ohm’s law.
22 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Sotirios A. Mallios and
Victor P. Pasko
Conclusions
I
The charge accumulation stage should be taken into account for the study
of the charge transfer to the ionosphere and to the ground in association
with lightning discharges. The amount of induced charge that is
transferred to the ionosphere during the charge accumulation phase prior
to the lightning is comparable to the charge that is transferred during the
slow transients after the lightning occurrence.
I
In addition to the charge accumulation and the lightning phases, the
quantitative description of the thunderstorm dissipation phase is also
important for understanding of the overall contribution of the
thunderstorms to the GEC.
I
Simulating a realistic thunderstorm based on the measurements of Mach
et al. [JGR, 114, D10204, 2009; JGR, 115, D03201, 2010; JGR, 116,
D05201, 2011] regarding the altitudes of the charge layers and the steady
state Wilson current that flows above the thundercloud to the ionosphere,
we were able to reproduce the dynamics of the thunderstorm. We showed
that the charge that is transfered to the ionosphere depends mainly on the
distance between the main positive and the main negative charge layers
and the deposited charge to the thundercloud. We also showed the
validity of our simple dipole model compared to the complex realistic
thunderstorm configuration.
FESD Meeting 2013
Introduction
Model Formulation
Results
Conclusions
23 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Conclusions (Cont)
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Results
I
The effects of the corona currents that flow from the ground to the bottom
of the thundercloud have been investigated. Corona currents tend to
decrease the amplitude of the electric field at the ground and thus, they
tend to decrease the amount of charge that is transfered to the ground.
The effects of the corona currents depend on the amplitude of the electric
field at the ground, which depends on the charge configuration of a given
thunderstorm.
I
A time dependent model has been presented for the study of the effect of
a single thunderstorm to the Global Electric Circuit. The results of this
model in the steady state are in agreement with a static model.
Conclusions
24 / 25
Time-Dependent
Modeling of Global
Electric Circuit
Processes
Sotirios A. Mallios and
Victor P. Pasko
FESD Meeting 2013
Introduction
Model Formulation
Results
Thank you for your attention!
Conclusions
This research was supported by NSF under grants AGS-0652148 and
AGS-1135446 to Penn State University.
25 / 25